Estimating beta-mixing coefficients via histograms

The literature on statistical learning for time series often assumes asymptotic independence or "mixing" of data sources. Beta-mixing has long been important in establishing the central limit theorem and invariance principle for stochastic processes; recent work has identified it as crucial to extending results from empirical processes and statistical learning theory to dependent data, with quantitative risk bounds involving the actual beta coefficients. There is, however, presently no way to actually estimate those coefficients from data; while general functional forms are known for some common classes of processes (Markov processes, ARMA models, etc.), specific coefficients are generally beyond calculation. We present an l1-risk consistent estimator for the beta-mixing coefficients, based on a single stationary sample path. Since mixing coefficients involve infinite-order dependence, we use an order-d Markov approximation. We prove high-probability concentration results for the Markov approximation and show that as d \rightarrow \infty, the Markov approximation converges to the true mixing coefficient. Our estimator is constructed using d dimensional histogram density estimates. Allowing asymptotics in the bandwidth as well as the dimension, we prove L1 concentration for the histogram as an intermediate step.